Subintuitionistic logics and their modal companions: a nested approach

In the present paper we deal with subintuitionistic logics and their modal companions. In particular, we introduce nested calculi for subintuitionistic systems and for modal logics in the (Formula presented.) modal cube ranging from (Formula presented.) to (Formula presented.). The latter calculi differ from standard nested systems, as there are multiple rules handling the modal operator. As an upshot, we get a purely syntactic proof of the Gödel-McKinsey-Tarski embedding which preserves the structure and the height of the derivations. Finally, we obtain a conservativity result for classical logic over a weak subintuitionistic system.